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Next part defective

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A machine is either in a good state, with defect probability $\frac{1}{5}$ per part, or in a worn state, with defect probability $\frac{3}{5}$ per part, and parts are independent given the state. The prior probabilities are $\frac{3}{4}$ for good and $\frac{1}{4}$ for worn. You inspect $2$ parts and find that exactly $1$ is defective. What is the probability that the next part is defective?

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Hint

Update the state probabilities first, then average the defect rate over them.

Solution

The chance of exactly $1$ defect in $2$ parts is $2 \times \frac{1}{5} \times \frac{4}{5} = \frac{8}{25}$ when good and $2 \times \frac{3}{5} \times \frac{2}{5} = \frac{12}{25}$ when worn.
The weights are $\frac{3}{4} \times \frac{8}{25} = \frac{6}{25}$ and $\frac{1}{4} \times \frac{12}{25} = \frac{3}{25}$ , so the posterior is $\frac{2}{3}$ good and $\frac{1}{3}$ worn.
The next part is defective with probability $\frac{2}{3} \times \frac{1}{5} + \frac{1}{3} \times \frac{3}{5} = \frac{2}{15} + \frac{3}{15} = \frac{1}{3}$ .

Answer: $\frac{1}{3}$

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